Disentangling orthogonal matrices

Area of Science:

• Computational Biology
• Applied Mathematics
• Structural Biology

Background:

• Cryo-electron microscopy (cryo-EM) relies on accurate molecular reconstruction.
• Solving linear systems with orthogonal matrices is crucial in various scientific domains.
• The orthogonal Procrustes problem is a related, well-studied problem.

Purpose of the Study:

• To address the challenge of solving linear systems with two unknown orthogonal matrices.
• To generalize existing methods for molecular reconstruction in cryo-EM.
• To introduce a novel algorithm with theoretical performance guarantees.

Main Methods:

• Development of a semi-definite programming (SDP) relaxation algorithm.
• Theoretical analysis of the algorithm's performance guarantees.
• Empirical validation and comparison with existing methods.

Main Results:

• The proposed SDP relaxation algorithm effectively solves linear systems with two unknown orthogonal matrices.
• The algorithm demonstrates superior performance compared to direct solving methods lacking orthogonal constraints.
• Theoretical guarantees support the algorithm's efficacy.

Conclusions:

• The novel SDP-based algorithm offers a significant advancement for molecular reconstruction in cryo-EM.
• This approach provides a robust solution for problems involving orthogonal matrices.
• The methodology is extendable to systems with more than two unknown orthogonal matrices.